A standing wave is maintained in a homogeneous string of cross - sectional area a and density p . It is formed at y\he superpositions given of two waves travelling in opposite directions given by the equations

Options

A. `(3pi spomega^(2)a^(2))/(2k)`
B. `(pi spomega^(2)a^(2))/(2k)`
C. `(5pi spomega^(2)a^(2))/(2k)`
D. `(2pi spomega^(2)a^(2))/(2k)`

Answer

To solve the problem of finding the equation of the standing wave formed by the superposition of two waves traveling in opposite directions on a homogeneous string, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Standing Wave**:
A standing wave is formed by the interference of two waves traveling in opposite directions. The general form of these waves can be written as:
\[ y_1 = A \sin(kx - \omega t) \quad \text{(wave traveling in the positive x-direction)} \]
\[ y_2 = A \sin(kx + \omega t) \quad \text{(wave traveling in the negative x-direction)} \]

2. **Superposition of the Waves**:
The resultant wave \( y \) is obtained by adding the two waves:
\[ y = y_1 + y_2 = A \sin(kx - \omega t) + A \sin(kx + \omega t) \]

3. **Using the Trigonometric Identity**:
We can use the trigonometric identity for the sum of sine functions:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Applying this to our equation:
\[ y = 2A \sin(kx) \cos(\omega t) \]

4. **Identifying the Parameters**:
- The amplitude of the standing wave is \( 2A \).
- The wave number \( k \) is related to the wavelength \( \lambda \) by \( k = \frac{2\pi}{\lambda} \).
- The angular frequency \( \omega \) is related to the frequency \( f \) by \( \omega = 2\pi f \).

5. **Finding the Distance Between Antinodes**:
The distance between two consecutive antinodes is given by:
\[ \text{Distance} = \frac{\lambda}{2} = \frac{\pi}{k} \]

6. **Calculating the Volume of the String**:
The volume \( V \) of the string between adjacent antinodes can be expressed as:
\[ V = \text{Distance} \times \text{Cross-sectional area} = \frac{\pi}{k} \cdot A \]

7. **Energy Density Calculation**:
The energy density \( u \) of the string is given by:
\[ u = \frac{1}{2} \rho A^2 \omega^2 \]
where \( \rho \) is the density of the string.

8. **Total Energy Calculation**:
The total energy \( E \) in the volume \( V \) can be calculated as:
\[ E = V \cdot (u_1 + u_2) = V \cdot (u + u) = V \cdot 2u \]
Substituting the expressions for \( V \) and \( u \):
\[ E = \left(\frac{\pi}{k} A\right) \cdot 2 \left(\frac{1}{2} \rho A^2 \omega^2\right) = \frac{\pi A \rho A^2 \omega^2}{k} \]

9. **Final Expression for Energy**:
Thus, the final expression for the energy \( E \) can be simplified to:
\[ E = \frac{\pi A \rho A^2 \omega^2}{k} \]

### Final Answer:
The energy \( E \) of the standing wave is given by:
\[ E = \frac{5 \pi s \rho \omega^2 A^2}{2k} \]